arXiv subjects

Mozhgan Mirzaei

Publications and source records attributed to Mozhgan Mirzaei.

6 recordsLinked to original sources

arXiv2021

Classes of ODE solutions: smoothness, covering numbers, implications for noisy function fitting, and the curse of smoothness phenomenon

Many numerical methods for recovering ODE solutions from data rely on approximating the solutions using basis functions or kernel functions under a least square criterion. The accuracy of this approach hinges on the smoothness of the solutions. This paper provides a theoretical foundation for these methods by establishing novel results on the smoothness and covering numbers of ODE solution classes (as a measure of their "size"). Our results provide answers to "how do the degree of smoothness and the "size" of a class of ODEs affect the "size" of the associated class of solutions?" We show that: (1) for $y^{'}=f\left(y\right)$ and $y^{'}=f\left(x,\,y\right)$, if the absolute values of all $k$th ($k\leqβ+1$) order derivatives of $f$ are bounded by $1$, then the solution can end up with the $(k+1)$th derivative whose magnitude grows factorially fast in $k$ -- "a curse of smoothness"; (2) our upper bounds for the covering numbers of the $(β+2)-$degree smooth solution classes are greater than those of the "standard" $(β+2)-$degree smooth class of univariate functions; (3) the mean squared error of least squares fitting for noisy recovery has a convergence rate no larger than $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ if $n=Ω\left(\left(β\sqrt{\log\left(β\vee1\right)}\right)^{4β+10}\right)$, and under this condition, the rate $\left(\frac{1}{n}\right)^{\frac{2\left(β+2\right)}{2\left(β+2\right)+1}}$ is minimax optimal in the case of $y^{'}=f\left(x,\,y\right)$; (4) more generally, for the higher order Picard type ODEs, $y^{\left(m\right)}=f\left(x,\,y,\,y^{'},\,...,y^{\left(m-1\right)}\right)$, the covering number of the solution class is bounded from above by the product of the covering number of the class $\mathcal{F}$ that $f$ ranges over and the covering number of the set where initial values lie.

math.ST↗

arXiv2020

A positive fraction mutually avoiding sets theorem

Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman showed that every set of $n$ points in the plane in general position contains a pair of mutually avoiding sets each of size at least $\sqrt{n/12}$. As a corollary, their result implies that for every set of $n$ points in the plane in general position one can find at least $\sqrt{n/12}$ segments, each joining two of the points, such that these segments are pairwise crossing. In this note, we prove a fractional version of their theorem: for every $k > 0$ there is a constant $\varepsilon_k > 0$ such that any sufficiently large point set $P$ in the plane contains $2k$ subsets $A_1,\ldots, A_{k},B_1,\ldots, B_k$, each of size at least $\varepsilon_k|P|$, such that every pair of sets $A = \{a_1,\ldots, a_k\}$ and $B = \{b_1,\ldots, b_k\}$, with $a_i \in A_i$ and $b_i \in B_i$, are mutually avoiding. Moreover, we show that $\varepsilon_k = Ω(1/k^4)$. Similar results are obtained in higher dimensions

math.CO↗

arXiv2020

On grids in point-line arrangements in the plane

The famous Szemerédi-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Turán-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $Ω({n^{1+\frac{1}{14}}})$ incidences.

math.CO↗

arXiv2019

Constructions of point-line arrangements in the plane with large girth

A classical result by Erdős, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a major open problem in extremal graph theory to decide if this bound is tight for all $k$. In this paper, we study the effect of forbidding short even cycles in incidence graphs of point-line arrangements in the plane. It is not known if the Erdős upper bound stated above can be improved to $o(n^{1+1/k})$ in this geometric setting, and in this note, we establish non-trivial lower bounds for this problem by modifying known constructions arising in finite geometries. In particular, by modifying a construction due to Labeznik and Ustimenko, we construct an arrangement of $n$ points and $n$ lines in the plane, such that their incidence graph has girth at least $k + 5$, and determines at least $Ω({n^{1+\frac{4}{k^2+6k-3}}})$ incidences. We also apply the same technique to Wenger graphs, which gives a better lower bound for $k=5.$

math.CO↗

arXiv2019

Exponential sum estimates over prime fields

In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.

math.NT↗

arXiv2019

A note on conditional expanders over prime fields

Let $\mathbb{F}_p$ be a prime field of order $p,$ and $A$ be a set in $\mathbb{F}_p$ with $|A| \leq p^{1/2}.$ In this note, we show that \[\max\{|A+A|, |f(A, A)|\}\gtrsim |A|^{\frac{6}{5}+\frac{4}{305}},\] where $f(x, y)$ is a non-degenerate quadratic polynomial in $\mathbb{F}_p[x, y].$ This improves a recent result given by Koh, Mojarrad, Pham, Valculescu (2018).

math.CO↗